This grant proposal is focused on developing new mathematical and statistical models to describe biological systems. Models to represent, help to understand, predict future behavior, and control biological systems are becoming more and more important and of widespread use in different fields related to biology and health care. Complex mathematical models are needed to model the complicated interactions between the physiological functions of biological systems, and to model the effect of interventions (e.g. therapy) on these functions. The specific aims of this grant focus on three areas of research. 1. Develop and investigate statistical models for biological population data. Biological data are always collected from some population of different individuals, and are often highly variable. This is mostly due to variability of physiological functions between individuals, and to measurement error. Statistical models are needed to deal with the complex structure of population data. I will (I) introduce a general methodology based on the use of sophisticated heteroscedastic statistical models, which does not explicitly formulate a model for interindividual variability but promises to be fast, efficient and unbiased; and (ii) investigate the performance of existing population models using realistic simulations including model misspecification. 2. Develop semi-mechanistic compartmental models. I focus on three main problems: (i) the development and investigation a new general class of compartmental pharmacokinetics"'pharmacodynamic (PK/PD) models, (ii) the development of semi-mechanistic black-box compartmental models to deal with non-linear PK systems, (iii) the development of the technology to apply well established semi-mechanistic linear black-box models to the purpose of PK control. 3. Develop new multivariate dynamic models. The main problem addressed is how to represent a system where multiple inputs (drugs) and multiple interrelated responses are measured. I propose different classes of models to do so based on spline networks and eventually neural networks. The proposed models can incorporate a compartmental sub-structure to easily deal with kinetics. Continuous and discrete time versions of the models are considered. The statistical and mathematical models introduced in the grant have widespread application to a variety of biological fields. However specific areas, directly linked to health care issues, are selected for active research and application of the proposed models. These areas correspond to experimental situations where the models proposed in the grant are particularly needed (nonlinear and multivariate dynamic), and represent continuations of already established collaborations with leading scientists. They include: computer control of ultra-short acting anaesthetic drugs administration, pharmacokinetics/pharmacodynamic of short-acting anesthetics, pharmacodynamic of nicotine and nicotine tolerance development, adenosine kinetics and metabolism and their relationship to adenosine pharmacodynamic effects, modeling of cardiovascular drugs effects on pharmacy dynamic responses (heart rate, blood pressure, and breathing variability) sampled at high rates.